Why is A cross B= ABsin(theta) false?

[Ashley1nOnly]

Homework Statement

problem 2.2(b)

Homework Equations

A cross B = ABsin(theta)

The Attempt at a Solution

I believe that the answer is that you cannot multiply two vectors but you can dot or cross them. In order to make this a true statement you have to take the magnitudes of A and B.

 

[DrClaude]

Note the change in font:
$$
\mathbf{A} \times \mathbf{B} = A B \sin \theta
$$
The author is using a common convention that $A = | \mathbf{A} |$, so it is the magnitude that is considered.

 

[Buzz Bloom]

Hi Ashley:

I think you will find this helpful.

https://en.wikipedia.org/wiki/Cross_product​

Regards,
Buzz

 

[Ashley1nOnly]

From my readings
When you cross two vectors you get a new vector that is orthogonal to them and that the vectors form a parallelogram.
The new vector c pointing perpendicular to them gives us the height. But the height is defined by by C(sin(theta)).
In order to find the area its Height times base. The length of the base is |B| and the height is |A|(sin(theta)). It's sin(theta) because we broke up the A vector into its two components and the height of vector A is opposite the angle which gives us sin.To continue and apply this knowledge

If we want to find the volume it's height times the area of the parallelogram. Now I know the height of the new vector C is Csin(theta) so now I just multiply that by the area I got from above. Is this right?

 

[Ashley1nOnly]

Plus A cross B is a vector and we should get another vector back. ABsin(theta) is not a vector so it would make the statement false. We have a vector equal to a scalar in this equation.

 

[DrClaude]

Plus A cross B is a vector and we should get another vector back. ABsin(theta) is not a vector so it would make the statement false. We have a vector equal to a scalar in this equation.

That's it.

But you see that the equation is almost correct.